Magneto-optical Kerr effect in spin split two-dimensional massive Dirac materials
MMagneto-optical Kerr effect in spin split two-dimensional massive Dirac materials
G. Catarina , , ∗ , N. M. R. Peres , , and J. Fern´andez-Rossier , † QuantaLab, International Iberian Nanotechnology Laboratory (INL), 4715-330 Braga, Portugal Centro de F´ısica das Universidades do Minho e do Porto and Departamento de F´ısica and QuantaLab,Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal (Dated: April 13, 2020)Two-dimensional (2D) massive Dirac electrons possess a finite Berry curvature, with Chern num-ber ± /
2, that entails both a quantized dc Hall response and a subgap full-quarter Kerr rotation.The observation of these effects in 2D massive Dirac materials such as gapped graphene, hexagonalboron nitride or transition metal dichalcogenides (TMDs) is obscured by the fact that Dirac conescome in pairs with opposite sign Berry curvatures, leading to a vanishing Chern number. Here, weshow that the presence of spin-orbit interactions, combined with an exchange spin splitting inducedeither by diluted magnetic impurities or by proximity to a ferromagnetic insulator, gives origin toa net magneto-optical Kerr effect in such systems. We focus on the case of TMD monolayers andstudy the dependence of Kerr rotation on frequency and exchange spin splitting. The role of thesubstrate is included in the theory and found to critically affect the results. Our calculations indi-cate that state-of-the-art magneto-optical Kerr spectroscopy can detect a single magnetic impurityin diluted magnetic TMDs.
I. INTRODUCTION
The electronic states of a two-dimensional (2D) gappedDirac Hamiltonian have a finite Berry curvature, withChern number C = ± /
2, that leads to a quantizeddc Hall conductivity. At finite frequencies, the Hall re-sponse is also peculiar and gives origin to a giant low-frequency Kerr rotation in thin-film topological insula-tors . The observation of these anomalous phenomenarequires a material realization of a massive Dirac elec-tron gas in two dimensions. Possible candidates are thesurfaces of three-dimensional topological insulators, thathost 2D massless Dirac cone states at the Γ point of theBrillouin zone . Both magnetic doping or spin proxim-ity effect can be used to open up a gap, which wouldpermit to probe the anomalous Hall response associatedto massive Dirac electrons in two dimensions.In this work, we explore an alternative route to un-veil the anomalous Hall response of 2D massive Diracmaterials. For that matter, we consider a differentclass of physical systems with strong light-matter cou-pling : the widely studied semiconducting transitionmetal dichalcogenide (TMD) monolayers, such as MoS .The low-energy electronic properties of these materialsare governed by states in the neighborhood of two non-equivalent valleys, which can be described in terms of aspin-valley coupled massive Dirac equation . In the pres-ence of time-reversal symmetry, the total Berry curvaturevanishes due to a perfect cancellation of the contributionscoming from the two valleys. However, the introduc-tion of an exchange spin splitting —which breaks time-reversal symmetry—, combined with the strong spin- ∗ [email protected] † On leave from Departamento de F´ısica Aplicada, Universidad deAlicante, 03690 San Vicente del Raspeig, Spain. orbit interactions , offsets this cancellation, leadingto an anomalous Hall response that results in a non-vanishing magneto-optical Kerr effect.We consider two different mechanisms to induce ex-change spin splitting in TMDs. These entail interactionof the electronic states in the valence and conductionbands of the TMD with magnetic atoms located either atthe TMD itself, as magnetic dopants in a diluted magneticsemiconductor (Fig. 1a), or at an adjacent ferromagneticinsulator, in which case exchange arises from spin prox-imity (Fig. 1b). Diluted magnetic doping of TMDs hasbeen considered theoretically and realized experi-mentally . Some intrinsic point defects in TMDs areexpected to be spin polarized , so that they can alsoact as magnetic centers. Exchange-driven spin splittingsin TMDs caused by proximity to ferromagnetic insulatorshave been reported both in experiments and in first-principle calculations . It has also been predictedthat an antiferromagnetic layered substrate can induce,by proximity effect, a spin splitting of TMD bands ; thisoccurs due to exchange interactions between the TMDand the surface layer of the substrate, which has ferro-magnetic order. II. MODEL HAMILTONIAN
We model TMD monolayers with the additional ex-change spin splitting through an Hamiltonian with threeterms, H = H MD + H soc + H ex . (1)The first two terms represent the well-known spin-valleycoupled massive Dirac model for TMDs in the trigonalprismatic configuration. These describe the low-energyelectronic properties of TMDs, which are governed bystates in the neighborhood of two non-equivalent pointsof the Brillouin zone: the so-called K and K (cid:48) valleys. The a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r (c)(a) (b) FIG. 1: (Color online) (a,b) Representation of the physicalsystem: monolayer transition metal dichalcogenide (TMD)in the presence of exchange interactions induced by dilutedmagnetic dopants (a) or by proximity effect to a ferromagneticinsulator (b). (c) Left panel: low-energy bands of monolayerTMDs. Blue/red lines stand for bands with spin up/downprojections, split due to spin-orbit interactions with differentmagnitude in the conduction and valence bands, λ v (cid:29) | λ c | .Solid/dashed lines represent Dirac bands obtained around the K / K (cid:48) valley. Right panel: effect of exchange at the bottom ofthe conduction bands and at the top of the valence bands. Acombination of both spin-orbit coupling and band-dependentexchange J c (cid:54) = J v leads to four non-degenerate effective bandgaps. first term corresponds to a massive Dirac Hamiltonian, H MD = (cid:126) v F ( k x τ z σ + k y σ ) + ∆2 σ , (2)where (cid:126) is the reduced Planck constant, v F is the Fermivelocity, k = ( k x , k y ) is the electron wave vector, τ z isthe valley operator with eigenvalues τ = ± (+ for K and − for K (cid:48) ), σ i ( i = 1 , ,
3) are Pauli matrices acting on thespace of the lowest-energy conduction and highest-energyvalence states, and ∆ is the bare band gap (or mass, inthe language of relativistic quantum mechanics). Thesecond term accounts for the strong spin-orbit couplingin TMDs , reading as H soc = λ c ( + σ ) + λ v ( − σ )2 τ z s z , (3)where 2 λ c/v is the spin splitting in the conduc-tion/valence bands and s z is the Pauli matrix for the out-of-plane spin component with eigenvalues s = +( ↑ ) , − ( ↓ ).On account of the different atomic orbital breakdown ofconduction and valence states, λ v (cid:29) | λ c | is verified formost of the TMD materials . Importantly, spin-orbitinteractions preserve time-reversal symmetry due to the so-called spin-valley coupling: states with spin ↑ in valley K have a Kramers partner in valley K (cid:48) with spin ↓ .The third term in Eq. (1), given by H ex = J c ( + σ ) + J v ( − σ )2 s z , (4)describes an exchange-driven spin splitting of 2 J c/v inthe conduction/valence bands. This term breaks time-reversal symmetry. In the case of diluted magnetic semi-conductors, it can be derived (see Appendix A) applyingfirst-order perturbation theory to a Kondo model withinthe so-called virtual crystal approximation . This sets J c/v = x imp (cid:104) M z (cid:105) γ c/v , (5)where x imp is the atomic ratio of magnetic impuri-ties, (cid:104) M z (cid:105) is their statistical average spin (local spinsare treated classically, within mean-field, and assumedto have orientation along z ), and γ c/v is a material-dependent parameter, formally defined as the expecta-tion value of the Kondo exchange coupling within con-duction/valence states. For conventional diluted mag-netic semiconductors based on II-VI compounds dopedwith Mn, experimental measurements yield values of γ c and γ v with opposite signs and magnitudes up to 1 eV .Therefore, a net exchange spin splitting ∆ J = J c − J v inthe order of tens of meV could be reached for Mn con-centrations of few percent. In the case of TMDs on topof ferromagnetic insulators, Eq. (4) has been used to account for giant band-dependent exchange spin split-tings predicted by first-principle calculations .Altogether, the model Hamiltonian described byEq. (1) can be considered as four independent copies (twoper spin and valley) of a massive Dirac model, each ofwhich with an effective gap given by E τ,s gap = ∆ + τ s ∆ λ + s ∆ J , (6)where ∆ λ = λ c − λ v . The corresponding band spectrum isdepicted in Fig. 1c. For J c = J v = 0, TMD bands appearas two spin-valley coupled Kramers doublets, split by thestrong spin-orbit interactions. As J c and J v are rampedup, spin-valley coupling is broken. For ∆ J (cid:54) = 0, we obtainfour non-degenerate effective gaps.Zeeman splitting, in the order of 0 . − , isignored in our model since, for most practical cases, it isdominated by the exchange term. III. MAGNETO-OPTICAL KERR EFFECT
We are interested in the magneto-optical Kerr responseof exchange spin split TMD monolayers. Specifically, wecompute the so-called Kerr rotation. When linearly po-larized light is shined into a magnetic/magnetized ma-terial, the reflected beam is in general elliptically polar-ized with its plane of polarization rotated with respectto that of the incident beam. The Kerr rotation is theangle of rotation of the plane of polarization. State-of-the-art experimental setups have reported Kerr rotationmeasurements with 10 nrad resolution .In Appendix B 1, we derive the equation that relatesthe complex Kerr angle and the optical conductivity of a2D material, θ K + i γ K (cid:39) πα σ xy σ (cid:16) πα σ xx σ + √ ε r (cid:17) + (cid:16) πα σ xy σ + i (cid:17) . (7)Here, θ K is the Kerr rotation, γ K is the Kerr ellipticity, α (cid:39) /
137 is the fine-structure constant, σ = e / (4 (cid:126) )is the universal conductivity of graphene ( e is the ele-mentary charge), ε r is the relative permittivity of thesubstrate on which the 2D material is deposited and σ xx ( σ xy ) stands for the longitudinal (Hall) component of the2D optical conductivity tensor. The derivation of this ex-pression assumes normal incidence (as usual within thepolar geometry) of linearly polarized light onto a 2D sys-tem with σ xx = σ yy and σ xy = − σ yx , placed on top ofa semi-infinite dielectric substrate. Moreover, it is onlyvalid in the limit of small θ K and γ K (see Eq. (B11) inAppendix B 1 for the general formula). It must be notedthat this equation is applicable for strictly 2D models asit depends on the 2D optical conductivity tensor, whoseunits are Siemens (instead of Siemens per meter). In Ap-pendix B 2, we show that Eq. (7) can be retrieved con-sidering multiple reflections in a stratified medium madeof a three-dimensional material encapsulated between airand a substrate, taking the limit d →
0, where d is thethickness of the material.In order to compute the 2D optical conductivity of ex-change spin split TMDs, Kubo formula is employed. Foreither ∆ λ = 0 or ∆ J = 0, straightforward calculationsyield σ xy = 0, which implies a vanishing magneto-opticalKerr response. Therefore, we conclude that, consideringour model Hamiltonian, magneto-optical Kerr effects areonly obtained in the presence of both spin-orbit interac-tions and band-dependent exchange spin splittings.In what follows, we focus on the case where the Fermilevel lies inside the gap, for which the results are inde-pendent of the temperature. In this regime, we can ig-nore the Landau level structure caused by an out-of-planemagnetic field given that, on its own, it does not lead toa finite Hall response . It must be noted, however, thatthe single-particle description followed here does not ac-count for the strong excitonic effects present in TMDs atcharge neutrality . This subject is left for a companionpublication .Eq. (7) shows that the complex Kerr angle has a non-trivial dependence both on the properties of the 2D ma-terial, encoded in its optical conductivity tensor, and onthe dielectric constant of the substrate. In this work,we consider substrates with ε r (cid:38)
2, which is in princi-ple the most natural case in experiments. Within thisassumption, we systematically find that, provided intra-band transitions are Pauli blocked at charge neutrality, Eq. (7) can be simplified into θ K + i γ K (cid:39) παε r − σ xy σ . (8)Thus, we see that Kerr rotation is governed by the realpart of σ xy .The optical conductivity of our model Hamiltonian canalways be expressed as a sum of spin- and valley-resolvedcontributions. In the limit where Eq. (8) is valid, we canalso define a spin- and valley-resolved Kerr rotation suchthat θ K ( ω ) = (cid:88) τ,s θ τ,s K ( ω ) , (9)where the dependence on ω , the angular frequency ofthe incident light, is explicitly indicated. Using Kuboformula, we get θ τ,s K ( ω ) = − τ αε r − E τ,s gap (cid:126) ω log (cid:12)(cid:12)(cid:12)(cid:12) (cid:126) ω + E τ,s gap (cid:126) ω − E τ,s gap (cid:12)(cid:12)(cid:12)(cid:12) , (10)where, for simplicity, we have not included a finite em-pirical broadening Γ within the Kubo formalism.The above equation describes the Kerr rotation asso-ciated to a single massive Dirac cone, with effective gap E τ,s gap , assuming charge neutrality and a dielectric sub-strate with ε r (cid:38)
2. It must be noted that the line shapeof ( ε r − θ τ,s K (cid:16) (cid:126) ωE τ,s gap (cid:17) is independent of any parameterof the theory. Taking ∆ λ = 0 and ∆ J = 0, the effectivegap becomes spin- and valley-independent and, summingover τ and s , we obtain a vanishing Kerr rotation due toopposite sign contributions at the two valleys (Fig. 2a).For both ∆ λ (cid:54) = 0 and ∆ J (cid:54) = 0, the presence of four non-degenerate effective gaps offsets this cancellation, leadingto a net Kerr rotation , as we show in Fig. 2b.In the dc limit, Eq. (10) gives θ τ K ( ω →
0) = − τ αε r − . (11)For frequencies below the gap, Eq. (10) approaches thedc limit rapidly, leading to nearly flat low-frequencyplateaus, as shown in Fig. 2a. As a side note, we stressthat, if we take ε r = 1 and consider a model Hamil-tonian with a single massive Dirac cone, Eq. (7) is nolonger valid, as the corresponding Kerr rotation is notsmall. Indeed, using the general formula derived in Ap-pendix B 1 (Eq. (B11)), we obtain subgap full-quarterplateaus, θ τ K ( ω < E gap ) (cid:39) − τ π/
2. Thus, we see that theinclusion of a substrate in the theory can significantlyaffect the results.We now address the properties of the (net) Kerr ro-tation in exchange spin split TMDs. Taking the low-frequency limit of Eqs. (9) and (10), we obtain θ K ( ω (cid:28) ∆) (cid:39) − αε r − (cid:18) (cid:126) ω ∆ (cid:19) ∆ J ∆ λ ∆ , (12) (a) (b) FIG. 2: (Color online) (a) Valley-resolved Kerr rotation θ τ K , as a function of the photon energy (cid:126) ω , for a two-dimensional massiveDirac material, with gap E gap , at charge neutrality and placed on top of a dielectric substrate with relative permittivity ε r (cid:38) ± α/ ( ε r − α is the fine-structure constant. (b)Kerr rotation as a function of the photon energy for monolayer MoS on top of SiO ( ε r = 2 . ), at charge neutrality and witha net exchange spin splitting ∆ J = 50 meV. Parameters: MoS bare band gap ∆ = 1 .
66 eV and spin-orbit coupling splittings2 λ c = − λ v = 148 meV ; empirical broadening Γ = 4 meV . A combination of both λ c (cid:54) = λ v and ∆ J (cid:54) = 0 leads tofour non-degenerate effective band gaps —one for each spin and valley— that are represented by the vertical lines, followingthe same color and dashing style as in Fig. 1b. The net Kerr response can be seen as a sum of spin- and valley-resolvedcontributions of massive Dirac electrons (a) that are offset in energy and thus do not cancel out. where we have also assumed ∆ (cid:29) ∆ J , ∆ λ . At higherfrequencies, no simple analytical expression can be found.This can be understood by the fact that the Kerr rotationis the sum of four curves (one per spin and valley) thathave a resonant peak at the absorption thresholds definedby the corresponding effective gap, as shown in Fig. 2b.Eq. (12) shows that, in the low-frequency regime, Kerrrotation varies linearly with ∆ J and thereby with theaverage magnetization of the impurities (in the case ofdiluted magnetic TMD semiconductors), by virtue ofEq. (5). For frequencies close to the absorption thresh-olds, the linearity breaks above a given value of ∆ J , aswe show in Fig. 3a. This is explained by the fact that,as we vary ∆ J , absorption thresholds are shifted in away that they eventually cross the photon energy, lead-ing to a non-monotonous dependence. Within the linearregime, it is also evident from Fig. 3a that the slopes de-pend strongly on ω . The frequency dependence of theseslopes, defined as η ( ω ) = ∂θ K ( ω ) ∂ ∆ J (cid:12)(cid:12)(cid:12)(cid:12) ∆ J =0 , (13)is presented in Fig. 3b.Finally, we focus on the case of diluted magnetic TMDsemiconductors and estimate the limits of magnetic mo-ment detection through Kerr rotation measurements.Specifically, given an experimental setup that permitsto detect Kerr rotation with resolution θ resK , we addressthe question of what is the smallest number of impuri-ties that can be probed. We assume that we are in theregime where Kerr rotation scales linearly with exchange, θ K ( ω ) = η ( ω )∆ J . Using Eq. (5), we can thus write | η ( ω )x imp (cid:104) M z (cid:105) ( γ c − γ v ) | > θ resK . (14) Taking the Abbe diffraction limit, we consider a laserspot with area A spot = π (cid:0) λ (cid:1) , where λ is the wavelength of the light and NA is the numerical aperture ofthe laser. Assuming a maximum of one impurity per unitcell, the number of impurities probed by the laser spotcan be written as N imp = x imp A spot A u.c. , where A u.c. = √ a is the area of the unit cell ( a is the lattice parameter).With this, we get N imp > πλ a NA θ resK | η ( ω ) (cid:104) M z (cid:105) ( γ c − γ v ) | . (15)In order to give rough estimations, we take a = 3 . (cid:6) A (having MoS as reference), NA ∼ θ resK = 10 nrad , (cid:104) M z (cid:105) ∼ | γ c − γ v | ∼ (taking as referenceconventional diluted magnetic semiconductors). For lowfrequencies, η ( ω ) can be obtained analytically throughEq. (12). Replacing ∆ = 1 .
66 eV , ∆ λ = −
151 meV and ε r = 3 . (considering MoS on top of SiO andtaking the dc limit of its relative permittivity), we get N imp (cid:38) (cid:126) ω [eV]) . At higher frequencies, we use the re-sults of Fig. 3b to obtain η ( ω ). Following a conservativeapproach, we avoid resonances and set (cid:126) ω = 1 .
65 eV,for which η ( ω ) takes the value marked as η . This leadsto N imp (cid:38) .
1, showing that a single impurity can bedetected. It must be noted that magneto-optical effectshave been used for single spin detection . IV. DISCUSSION AND CONCLUSIONS
We have presented a theory for magneto-optical Kerreffects in 2D materials whose low-energy bands are de-scribed by a spin split massive Dirac equation. Using the (a) (b)
FIG. 3: (Color online) (a) Kerr rotation, as a function of the net exchange spin splitting, for monolayer MoS on top of SiO ,at charge neutrality, for two photon energies. Parameters as in Fig. 2b. Linear dependence is observed in the small ∆ J limit;the corresponding slopes, marked by the dashed lines, are verified to depend strongly on (cid:126) ω . For larger ∆ J , a non-monotonousdependence is obtained. The origin of this behavior is the shifting of absorption thresholds (as the ones marked by the verticallines in Fig. 2b) as ∆ J is ramped up, which causes resonant peaks to cross the photon energy. (b) Derivative of Kerr rotationwith respect to net exchange spin splitting ∆ J , evaluated at ∆ J = 0, as a function of the photon energy, for the same systemas in (a). Marked points correspond to the slopes shown in (a). standard Fresnel formalism, we have obtained the equa-tion that relates the complex Kerr angle with the opticalconductivity tensor of a 2D system, considering the effectof a substrate. We have found that a combination of bothspin-orbit interactions and band-dependent spin splittingin the model leads to an anomalous Hall conductivitythat gives origin to a non-vanishing magneto-optical Kerrresponse. We have focused our theory in transition metaldichalcogenide monolayers, for which spin-orbit interac-tions are strong, and considered an exchange spin split-ting induced either by diluted magnetic impurities or byproximity effects to a ferromagnetic insulator. Our for-malism can be extended to tight-binding Hamiltoniansand to other types of magnetic order .The main results, obtained at charge neutrality andfor substrates with relative permittivity ε r (cid:38)
2, are thefollowing. First, we have obtained a simplified expres-sion which shows that Kerr rotation is governed by thereal part of the Hall conductivity and therefore permitsto define spin- and valley-resolved contributions. Second,we have shown that a single valley of a 2D gapped Diracmodel (with gap E gap ) entails a Kerr rotation with op-posite sign for each of the valleys and whose frequencydependence is given by a function that depends only on ω/E gap , taking the value − τ αε r − in the dc limit, where τ = ± is the valley index and α is the fine-structureconstant. Third, we have seen that the model Hamil-tonian for exchange spin split TMDs can be consideredas four copies (two per spin and valley) of a gappedDirac equation with non-degenerate effective gaps, suchthat the (net) Kerr rotation can be interpreted as a non-cancellation of spin- and valley-resolved features of a 2Dmassive Dirac theory. Fourth, we have addressed the useof Kerr rotation measurements to probe magnetic mo-ments in diluted magnetic TMD semiconductors, show-ing that state-of-the-art experimental setups can detectsignal coming from a single impurity. The role of excitonic corrections will be the subject offuture work. Acknowledgments
We thank Allan H. MacDonald, Elaine Li, Ale-jandro Molina-S´anchez and Jo˜ao C. G. Henriques forfruitful discussions. G. C. acknowledges Funda¸c˜aopara a Ciˆencia e a Tecnologia (FCT) for GrantNo. SFRH/BD/138806/2018. G. C. and J. F.-R. acknowledge financial support from FCT throughGrant No. P2020-PTDC/FIS-NAN/4662/2014. N.M. R. P. acknowledges financial support from Euro-pean Commission through project “Graphene-DrivenRevolutions in ICT and Beyond” (Ref. No.785219), FCT in the framework of Strategic Financ-ing (Ref. No. UID/FIS/04650/2019), and COM-PETE2020, PORTUGAL2020, FEDER and FCT forGrants No. PTDC/FIS-NAN/3668/2013, No. POCI-01-0145-FEDER-028114, No. POCI-01-0145-FEDER-029265 and No. PTDC/NAN-OPT/29265/2017. J.F.-R. acknowledges FCT for Grant No. UTAP-EXPL/NTec/0046/2017, as well as Generalitat Valen-ciana funding Prometeo2017/139 and MINECO-Spain(Grant No. MAT2016-78625-C2).
Appendix A: Exchange spin splitting in dilutedmagnetic semiconductors
Following Ref. 34, we model the exchange interactionbetween band electrons and diluted magnetic impuritiesthrough a Kondo-like exchange term, V = (cid:88) R i J ( R i ) M ( R i ) · s , (A1)where M ( R i ) is the vector of Pauli operators for the spinof magnetic impurities located at positions R i , s is thevector of Pauli operators for the spin of band electronsand J ( R i ) are exchange coupling constants.Treating the local spins classically and within mean-field, we replace M ( R i ) by its statistical average (cid:104) M (cid:105) .In addition, we assume an average magnetization alongthe z direction. Moreover, we employ the so-calledvirtual crystal approximation, making (cid:80) R i J ( R i ) → x imp (cid:80) R J ( R ), where x imp is the atomic ratio of mag-netic impurities and R denotes the positions of latticesites. With this, we get V = s z (cid:104) M z (cid:105) x imp (cid:88) R J ( R ) . (A2)To first order in perturbation theory, Eq. (A2) leads toa correction of the energy levels given by δE (1) = s (cid:104) M z (cid:105) x imp (cid:104) ψ | (cid:88) R J ( R ) | ψ (cid:105) , (A3)where ψ is the wave function of the unperturbed Hamil-tonian, which is assumed to be diagonal in the subspaceof s z with eigenvalues s = +( ↑ ) , − ( ↓ ). We now noticethat the matrix element present in the above equationdepends on the atomic orbital breakdown of ψ , suchthat it can be different for electrons in distinct bands.Taking this into account, and considering the low-energyDirac model for TMD monolayers, we write δE (1) c/v = s (cid:104) M z (cid:105) x imp γ c/v (A4)where γ c/v is the matrix element of the exchange cou-pling constants for conduction/valence states. Finally,we define J c/v = (cid:104) M z (cid:105) x imp γ c/v , (A5)such that 2 J c/v is the exchange spin splitting in the con-duction/valence bands, as captured by Eq. (4). Appendix B: Magneto-optical Kerr effect intwo-dimensional systems1. Formalism
We consider a 2D system lying at the xy plane, withair above ( z >
0) and a substrate below ( z < ε r . Both media aretaken as semi-infinite, disregarding any phenomenon ofmultiple reflections in stratified media.We assume normal incidence of linearly polarizedmonochromatic light and write its electric field as E (i) ( z, t ) = E (i) x u x e i ( − ωc z − ωt ) , (B1)where t is the time, ω is the angular frequency of the lightand c is the speed of light in vacuum. The electric field of the reflected and the transmitted light can be writtenas E (r) ( z, t ) = (cid:16) E (r) x u x + E (r) y u y (cid:17) e i ( ωc z − ωt ) (B2)and E (t) ( z, t ) = (cid:16) E (t) x u x + E (t) y u y (cid:17) e i ( −√ ε r ωc z − ωt ) , (B3)respectively. The corresponding magnetic fields B (i) , B (r) and B (t) are obtained via Maxwell’s equations.The interface conditions at z = 0 impose u z × (cid:104) E (i) (0 , t ) + E (r) (0 , t ) − E (t) (0 , t ) (cid:105) = , (B4) u z × (cid:104) B (i) (0 , t ) + B (r) (0 , t ) − B (t) (0 , t ) (cid:105) = µ j s , (B5)where µ is the vacuum permittivity and j s is the surfacecurrent density at the z = 0 plane. Applying Ohm’s law,we write j s = (cid:18) σ xx σ xy σ yx σ yy (cid:19) · (cid:32) E (t) x (0 , t ) E (t) y (0 , t ) (cid:33) , (B6)where σ ab ( a, b = x, y ) are the components of the op-tical conductivity tensor of the 2D material. Assumingthat σ xx = σ yy and σ xy = − σ yx , straightforward manip-ulation permits to obtain the reflection coefficients forright/left-handed light as r ± = E (r) ± E (i) x / √ − √ ε r − cµ σ ∓ √ ε r + cµ σ ∓ , (B7)where √ E (r) ± = E (r) x ± i E (r) y and σ ± = σ xx ± i σ xy .On the other hand, we can also write r + r − = (cid:12)(cid:12)(cid:12) E (r)+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (r) − (cid:12)(cid:12)(cid:12) e i( φ + − φ − ) , (B8)where E (r) ± = (cid:12)(cid:12)(cid:12) E (r) ± (cid:12)(cid:12)(cid:12) e i φ ± . We now identify the Kerr rota-tion as θ K = φ − − φ + γ K throughtan γ K = (cid:12)(cid:12)(cid:12) E (r)+ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) E (r) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (r)+ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E (r) − (cid:12)(cid:12)(cid:12) . (B10)As final result, we gettan (cid:16) γ K + π (cid:17) e − i2 θ K = (cid:18) − √ ε r − cµ σ − √ ε r + cµ σ − (cid:19) (cid:18) √ ε r + cµ σ + − √ ε r − cµ σ + (cid:19) , (B11)which is the general formula for magneto-optical Kerreffect in 2D systems. The above equation shows thata non-vanishing magneto-optical Kerr response implies σ + (cid:54) = σ − , which in turn implies σ xy (cid:54) = 0.In the limit of small θ K and γ K , we writetan (cid:0) γ K + π (cid:1) e − i2 θ K (cid:39) − i2 θ K + 2 γ K and Eq. (B11) issimplified into Eq. (7).
2. Agreement with the three-dimensional case
We now consider a stratified medium made of a mag-netic material with thickness d , encapsulated betweenair ( z >
0) and a substrate ( z < − d ). As in Section B 1,we treat air as vacuum and assume a non-magnetic di-electric substrate with relative permittivity ε r . Both airand the substrate are taken as semi-infinite but, in con-trast to the previous derivation, the finite thickness ofthe magnetic medium obliges us to account for multiplereflections within the Fresnel formalism. Regarding theproperties of the magnetic material, we assume that itspermittivy tensor can be expressed as ε MM = ε ε xx ε xy − ε xy ε xx
00 0 ε zz , (B12)where ε is the vacuum permittivity.Similarly to what was done in the previous section, weobtain the reflection coefficients by imposing the interfaceconditions at z = 0 and z = − d . The major differenceis that, in the magnetic medium, the propagation of thelight is not isotropic. Indeed, the electric field of thelight has components along the u ± = ( u x ± i u y ) / √ n ± = (cid:112) ε xx ± i ε xy . In addition, surface currents are now disregarded. After some straightforward algebra, we get r ± = 1 − h ( n ∓ )1 + h ( n ∓ ) , (B13)with h ( n ± ) = n ± f ( n ± ) − g ( n ± ) f ( n ± ) + g ( n ± ) , (B14) f ( n ± ) = ( n ± + √ ε r ) e − i ωc n ± d , (B15) g ( n ± ) = ( n ± − √ ε r ) e i ωc n ± d , (B16)which is the general formula in the three-dimensionalcase.In the limit of d →
0, we relate the optical conductivityin two and three dimensions via σ = σ d . (B17)Using the general (tensor) relation ε = ε + i σ ω (B18)and expanding equation Eq. (B13) to leading order in d ,we obtain r ± = 1 − √ ε r − cµ σ ∓ √ ε r + cµ σ ∓ , (B19)thus recovering Eq. (B7). D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den Nijs, Phys. Rev. Lett. , 405 (1982). W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. ,057401 (2010). T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He,L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K.Xue, Phys. Rev. Lett. , 266803 (2009). R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang,X. Dai, and Z. Fang, Science , 61 (2010),https://science.sciencemag.org/content/329/5987/61.full.pdf. F. Katmis, V. Lauter, F. S. Nogueira, B. A. Assaf, M. E.Jamer, P. Wei, B. Satpati, J. W. Freeland, I. Eremin, D. Heiman, P. Jarillo-Herrero, and J. S. Moodera, Na-ture , 513 (2016). K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,Phys. Rev. Lett. , 136805 (2010). A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim,C.-Y. Chim, G. Galli, and F. Wang, NanoLetters , 1271 (2010), pMID: 20229981,https://doi.org/10.1021/nl903868w . Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,and M. S. Strano, Nature Nanotechnology , 699 (2012). D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys.Rev. Lett. , 196802 (2012). G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao,
Phys. Rev. B , 085433 (2013). K. Ko´smider, J. W. Gonz´alez, and J. Fern´andez-Rossier,Phys. Rev. B , 245436 (2013). H. Da, L. Gao, W. Ding, and X. Yan, The Journal of Phys-ical Chemistry Letters , 3805 (2017), pMID: 28766341,https://doi.org/10.1021/acs.jpclett.7b01786 . Y. C. Cheng, Z. Y. Zhu, W. B. Mi, Z. B. Guo, andU. Schwingenschl¨ogl, Phys. Rev. B , 100401 (2013). A. Ramasubramaniam and D. Naveh, Phys. Rev. B ,195201 (2013). R. Mishra, W. Zhou, S. J. Pennycook, S. T. Pantelides,and J.-C. Idrobo, Phys. Rev. B , 144409 (2013). Y. C. Cheng, Q. Y. Zhang, and U. Schwingenschl¨ogl, Phys.Rev. B , 155429 (2014). A. N. Andriotis and M. Menon, Phys. Rev. B , 125304(2014). J. Wang, F. Sun, S. Yang, Y. Li, C. Zhao, M. Xu, Y. Zhang,and H. Zeng, Applied Physics Letters , 092401 (2016),https://doi.org/10.1063/1.4961883 . B. Xia, Y. Yang, J. Ma, K. Tao, and D. Gao, AppliedPhysics Express , 093002 (2017). M. T. Dau, C. Vergnaud, M. Gay, C. J. Alvarez,A. Marty, C. Beign´e, D. Jalabert, J.-F. Jacquot, O. Re-nault, H. Okuno, and M. Jamet, APL Materials , 051111(2019), https://doi.org/10.1063/1.5093384 . J. Hong, Z. Hu, M. Probert, K. Li, D. Lv, X. Yang, L. Gu,N. Mao, Q. Feng, L. Xie, J. Zhang, D. Wu, Z. Zhang,C. Jin, W. Ji, X. Zhang, J. Yuan, and Z. Zhang, NatureCommunications , 6293 (2015). W.-F. Li, C. Fang, and M. A. van Huis, Phys. Rev. B ,195425 (2016). M. A. Khan, M. Erementchouk, J. Hendrickson, and M. N.Leuenberger, Phys. Rev. B , 245435 (2017). M. A. Khan and M. N. Leuenberger, Journal of Physics:Condensed Matter , 155802 (2018). C. Zhao, T. Norden, P. Zhang, P. Zhao, Y. Cheng,F. Sun, J. P. Parry, P. Taheri, J. Wang, Y. Yang,T. Scrace, K. Kang, S. Yang, G.-x. Miao, R. Sabirianov,G. Kioseoglou, W. Huang, A. Petrou, and H. Zeng, NatureNanotechnology , 757 (2017). D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng, N. Sivadas,B. Huang, E. Schmidgall, T. Taniguchi, K. Watanabe,M. A. McGuire, W. Yao, D. Xiao, K.-M. C. Fu, and X. Xu,Science Advances (2017), 10.1126/sciadv.1603113,https://advances.sciencemag.org/content/3/5/e1603113.full.pdf. K. L. Seyler, D. Zhong, B. Huang, X. Linpeng,N. P. Wilson, T. Taniguchi, K. Watanabe, W. Yao,D. Xiao, M. A. McGuire, K.-M. C. Fu, andX. Xu, Nano Letters , 3823 (2018), pMID: 29756784,https://doi.org/10.1021/acs.nanolett.8b01105 . T. Norden, C. Zhao, P. Zhang, R. Sabirianov, A. Petrou,and H. Zeng, Nature Communications , 4163 (2019). J. Qi, X. Li, Q. Niu, and J. Feng, Phys. Rev. B , 121403(2015). Q. Zhang, S. A. Yang, W. Mi, Y. Cheng, andU. Schwingenschl¨ogl, Advanced Materials , 959 (2016),https://onlinelibrary.wiley.com/doi/pdf/10.1002/adma.201502585. N. Li, J. Zhang, Y. Xue, T. Zhou, and Z. Yang, Phys.Chem. Chem. Phys. , 3805 (2018). K. Zollner, P. E. Faria Junior, and J. Fabian, Phys. Rev.B , 085128 (2019). L. Xu, M. Yang, L. Shen, J. Zhou, T. Zhu, and Y. P. Feng, Phys. Rev. B , 041405 (2018). J. K. Furdyna, Journal of Applied Physics , R29 (1988),https://doi.org/10.1063/1.341700 . B. Scharf, G. Xu, A. Matos-Abiague, and I. ˇZuti´c, Phys.Rev. Lett. , 127403 (2017). Y. Li, J. Ludwig, T. Low, A. Chernikov, X. Cui, G. Arefe,Y. D. Kim, A. M. van der Zande, A. Rigosi, H. M. Hill,S. H. Kim, J. Hone, Z. Li, D. Smirnov, and T. F. Heinz,Phys. Rev. Lett. , 266804 (2014). D. MacNeill, C. Heikes, K. F. Mak, Z. Anderson,A. Korm´anyos, V. Z´olyomi, J. Park, and D. C. Ralph,Phys. Rev. Lett. , 037401 (2015). G. Aivazian, Z. Gong, A. M. Jones, R.-L. Chu, J. Yan,D. G. Mandrus, C. Zhang, D. Cobden, W. Yao, and X. Xu,Nature Physics , 148 (2015). A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke,A. Kis, and A. Imamo˘glu, Nature Physics , 141 (2015). A. Kapitulnik, J. Xia, E. Schemm, and A. Palevski, NewJournal of Physics , 055060 (2009). C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.Louie, J. Xia, and X. Zhang, Nature , 265 (2017). G. Catarina, J. Have, J. Fern´andez-Rossier, and N. M. R.Peres, Phys. Rev. B , 125405 (2019). G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz,X. Marie, T. Amand, and B. Urbaszek, Rev. Mod. Phys. , 021001 (2018). J. C. G. Henriques, G. Catarina, A. T. Costa,J. Fern´andez-Rossier, and N. M. R. Peres, Phys. Rev.B , 045408 (2020). T. J. Constant, S. M. Hornett, D. E. Chang, andE. Hendry, Nature Physics , 124 (2016). O. A. Ajayi, J. V. Ardelean, G. D. Shepard, J. Wang,A. Antony, T. Taniguchi, K. Watanabe, T. F. Heinz,S. Strauf, X.-Y. Zhu, and J. C. Hone, 2D Materials ,031011 (2017). J. Berezovsky, M. H. Mikkelsen, O. Gy-wat, N. G. Stoltz, L. A. Coldren, andD. D. Awschalom, Science , 1916 (2006),https://science.sciencemag.org/content/314/5807/1916.full.pdf. M. Atat¨ure, C. Dreiser, A. Badolato, and A. Imamoglu,Nature Physics , 101 (2007). N. Sivadas, S. Okamoto, and D. Xiao, Phys. Rev. Lett.117